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Gujarat Technological UniversitySummer 2024 Examination

GTU 3110015 Mathematics - 2 (Maths 2) Summer 2024 Paper Solution & PDF

B.E. · Common Engineering · Semester 1 · Subject Code: 3110015
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours2:30 PM – 5:30 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)
Find curl of v⃗=(xyz)i^+(3x2y)j^+(xz2−y2z)k^\vec{v} = (xyz)\hat{i} + (3x^2 y)\hat{j} + (xz^2 - y^2 z)\hat{k} at (2,−1,1)(2, -1, 1).
3 Marks
(b)
If a force F⃗=2x2yi^+3xyj^\vec{F} = 2x^2 y\hat{i} + 3xy\hat{j} displaces a particle in the xyxy-plane from (0,0)(0, 0) to (1,4)(1, 4) along a curve y=4x2y = 4x^2. Find the work done.
4 Marks
(c)
State and apply Green’s theorem to evaluate ∫C[(2x2−y2) dx+(x2+y2) dy]\int_C \left[ (2x^2 - y^2) \, dx + (x^2 + y^2) \, dy \right] where CC is the boundary of the area enclosed by the xx-axis and the upper half of the circle x2+y2=a2x^2 + y^2 = a^2.
7 Marks

Question 2

14 MarksMedium
(a)
Find Laplace transform of f(t)=∫0tsin⁡tt dtf(t) = \int_0^t \frac{\sin t}{t} \, dt
3 Marks
(b)
Find the Fourier cosine integral of f(x)=e−kxf(x) = e^{-kx}, where x>0,k>0x > 0, k > 0.
4 Marks
(c)
State convolution theorem and use it to find inverse Laplace transform of 1(s2+a2)2\frac{1}{(s^2 + a^2)^2}
7 Marks
OR OPTION
(c)
Using Laplace transform solve the following initial value problem:

y′′+4y′+8y=1,y(0)=0,  y′(0)=1y'' + 4y' + 8y = 1, \quad y(0) = 0, \; y'(0) = 1

7 Marks

Question 3

14 MarksMedium
(a)
Solve d2xdt2+6dxdt+9x=0\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 9x = 0
3 Marks
(b)
Find the inverse Laplace transform of se−s2+πe−ss2+π2\frac{s e^{-\frac{s}{2}} + \pi e^{-s}}{s^2 + \pi^2}
4 Marks
(c)
Solve:
iy+px=x4p2y + px = x^4 p^2
iip2−xp+y=0p^2 - xp + y = 0
7 Marks
OR OPTION
(a)
Solve (y2−x2) dx+2xy dy=0(y^2 - x^2) \, dx + 2xy \, dy = 0.
3 Marks
(b)
Find the Laplace transform of the waveform f(t)=2t3,0≤t≤3f(t) = \frac{2t}{3}, \quad 0 \le t \le 3
4 Marks
(c)
Find the series solution of (1+x2)y′′+xy′−9y=0(1 + x^2)y'' + xy' - 9y = 0.
7 Marks

Question 4

14 MarksMedium
(a)
Solve 9yy′+4x=09yy' + 4x = 0.
3 Marks
(b)
If y1=xy_1 = x is one of the solution of x2y′′+xy′−y=0x^2 y'' + xy' - y = 0, find the second solution.
4 Marks
(c)
Using the method of variation of parameter, solve d2ydx2+y=sin⁡x\frac{d^2 y}{dx^2} + y = \sin x
7 Marks
OR OPTION
(a)
Find Laplace transform of t2u(t−2)t^2 u(t - 2).
3 Marks
(b)
Solve (D2+9)y=2sin⁡3x+cos⁡3x(D^2 + 9)y = 2\sin 3x + \cos 3x, where D=ddxD = \frac{d}{dx}.
4 Marks
(c)
Using the method of undetermined coefficients, solve y′′−2y′+5y=5x3−6x2+6xy'' - 2y' + 5y = 5x^3 - 6x^2 + 6x
7 Marks

Question 5

14 MarksMedium
(a)
Solve x2y′′−20y=0x^2 y'' - 20y = 0.
3 Marks
(b)
Solve (D2−1)y=xex(D^2 - 1)y = x e^x where D=ddxD = \frac{d}{dx}.
4 Marks
(c)
Using Frobenius method, solve 4xd2ydx2+2dydx+y=04x \frac{d^2 y}{dx^2} + 2 \frac{dy}{dx} + y = 0
7 Marks
OR OPTION
(a)
Classify the singular points of the equation x3(x−2)y′′+x3y′+6y=0x^3 (x - 2) y'' + x^3 y' + 6y = 0.
3 Marks
(b)
Prove that ddx[Jn2(x)]=x2n[Jn−12(x)−Jn+12(x)]\frac{d}{dx}[J_n^2 (x)] = \frac{x}{2n}[J_{n-1}^2 (x) - J_{n+1}^2 (x)]
4 Marks
(c)
Show that ∫−11x2Pn−1(x)Pn+1(x) dx=2n(n+1)(2n−1)(2n+1)(2n+3)\int_{-1}^1 x^2 P_{n-1}(x) P_{n+1}(x) \, dx = \frac{2n(n+1)}{(2n-1)(2n+1)(2n+3)}
7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Mathematics - 2 (Maths 2) (Summer 2024, B.E. · Common Engineering, Sem 1). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.

Transcribed for student exam preparation from Gujarat Technological University official examination archives. Questions, syllabus guidelines, and curriculum marking schemes remain the intellectual property of Gujarat Technological University.

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